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[parent] linear differential equation of first order (Derivation)

An ordinary linear differential equation of first order has the form

$\displaystyle \frac{dy}{dx}+P(x)y = Q(x),$ (1)

where $ y$ means the unknown function, $ P$ and $ Q$ are two known continuous functions.

For finding the solution of (1), we may seek a function $ y$ which is product of two functions:

$\displaystyle y(x) = u(x)v(x)$ (2)

One of these two can be chosen freely; the other is determined according to (1).

We substitute (2) and the derivative $ \frac{dy}{dx} = u\frac{dv}{dx}+v\frac{du}{dx}$ in (1), getting $ u\frac{dv}{dx}+v\frac{du}{dx}+Puv = Q$, or

$\displaystyle u\left(\frac{dv}{dx}+Pv\right)+v\frac{du}{dx} = Q.$ (3)

If we chose the function $ v$ such that
$\displaystyle \frac{dv}{dx}+Pv = 0,$
this condition may be written
$\displaystyle \frac{dv}{v} = -P\,dx.$
Integrating here both sides gives $ \ln{v} = -\int P\,dx$ or
$\displaystyle v = e^{-\int Pdx},$
where the exponent means an arbitrary antiderivative of $ -P$. Naturally, $ v(x) \neq 0$.

Considering the chosen property of $ v$ in (3), this equation can be written

$\displaystyle v\frac{du}{dx} = Q,$
i.e.
$\displaystyle \frac{du}{dx} = \frac{Q(x)}{v(x)},$
whence
$\displaystyle u = \int\frac{Q(x)}{v(x)}\,dx+C = C+\!\int Qe^{\int Pdx}dx.$

So we have obtained the solution

$\displaystyle y = e^{-\int P(x)dx}\left[C+\!\int Q(x)e^{\int P(x)dx}dx\right]$ (4)

of the given differential equation (1).

The result (4) presents the general solution of (1), since the arbitrary constant $ C$ may be always chosen so that any given initial condition

$\displaystyle y = y_0 \quad \mathrm{when}\quad x = x_0$
is fulfilled.



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See Also: separation of variables

Other names:  linear ordinary differential equation of first order

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Cross-references: initial condition, general solution, differential equation, equation, property, antiderivative, exponent, sides, derivative, product, solution, continuous functions, function
There are 4 references to this entry.

This is version 8 of linear differential equation of first order, born on 2007-01-05, modified 2007-01-28.
Object id is 8717, canonical name is LinearDifferentialEquationOfFirstOrder.
Accessed 2296 times total.

Classification:
AMS MSC: 34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general)
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